Permutation:
The theory of how hairdos evolved
The New Uxbridge Dictionary Brook-Taylor et.al.
VI.3. PERMUTATION GROUP DECOMPOSITIONS 103 Finding the automorphism group of an object can be hard. In many practically relevant cases, however the objects have substructures that need to be preserved by automorphisms. In this case we can relate automorphisms of the structure with automorphisms of the substructures in a meaningful way.
In this section we shall look at two kinds of decompositions that lend them-selves to nice group theoretic structures.
By labelling relevant parts of the object acted on by G we can assume without loss of generality that the automorphism group group on a set Ω.
We start with the situation that Ω can be partitioned into two subsets that must be both preserved as sets, that is Ω = ∆ ∪ Λ with ∆ ∩ Λ = ∅. This could be distinct objects — say vertices and edges of a graph — or objects that structurally cannot be mapped to each other – for example verticed of a graph of different degree. (The partition could be into more than two cells, in which case one could take a union of orbits and iterate, taking the case of two as base case.)
In this case G must permute the elements of ∆ amongst themselves, as well as the elements of Λ. We thus can write every element of G as a product of a tation on ∆ and a permutation on Λ. Since the sets are disjoint, these two permu-tations commute.
Group theoretically, this means that we get a homomorphism φ∶ G → S∆, as well as a homomorphism ψ∶ G → SΛ. The intersection Kern φ ∩ Kern ψ is clearly trivial, as elements of G are determined uniquely by their action on Ω = ∆ ∪ Λ. We thus can combine these maps to an embedding (an injective group homomorphisms)
ι∶ G → S∆× SΛ, g ↦ (φ(g), ψ(g))
If we consider the direct product as acting on the disjoint union of ∆ and Λ, this map is simply the identity map on permutations.
The images of φ and ψ are typically not the whole symmetric groups. Thus, setting A = φ(G) and B = ψ(G), we can consider G as a subgroup of A × B, and call it a subdirect product of A and B. All intransitive permutation groups are such subdirect products.
Note that a subdirect product does not need to be a direct product itself, but can be a proper subgroup, see Exercise ??. There is a description on possible groups arising this way, based on isomorphic factor groups. Doing this however requires a bit more of group theory than we are prepared to use here.
We also note that it is not hard to classify subdirect products abstractly, and that this indeed has been done to enumerate intransitive permutation groups.
Block systems and Wreath Products
The second kind of decomposition we shall investigate is the case of a group G that is transitive on Ω, but permutes a partition into subsets (which then need to be of equal size).
Figure VI.7: Two graphs with an imprimitive automorphism group.
For example, in the two graphs in Figure VI.7, one such partition would be giv-en by vertices that are diagonally opposed. Neither diagonal is fixed, but a diagonal must be mapped to a diagonal again.
In the graph on the right, constructed from four triangles that are connected in all possible ways with the two neighbors, such a partition would be given by the triangles.
We formalize this in the following definition:
Definition VI.11: Let G act transitively on Ω. A block system (or system of imprim-itivity) for G on Ω is a partition B of Ω that is invariant under the action of G.
Example: For example, taking G = GLn(F) for a finite field F, acting on the nonzero vectors of Fn, the sect of vectors that span the same 1-dimensional space, that is those that are equivalent under multiplication by F∗, form a block system.
We note a few basic properties of block systems, the proof of which is left as an exercise.
Lemma VI.12: Let G act transitively on Ω with ∣Ω∣ = n.
a) There are two trivial block systems, B0= {{ω}}ω∈Ω, as well as B∞= {Ω}.
b) All blocks in a block system must have the same size.
c) If B is a block system, consisting of a blocks of size b, then n = ab.
d) If B is a block system, then G acts transitively on the blocks in B.
e) A block system is determined uniquely by one of its blocks.
We note — See section ?? — that part a) of the lemma is the best possible – there are groups which only afford the trivial block systems.
A connection between blocks and group structure is given by the following proposition which should be seen as an extension of Theorem VI.5:
Proposition VI.13: Let G act transitively on Ω and let ω ∈ Ω with S = StabG(ω).
There is a bijection between block systems of G on Ω and subgroups S ≤ U ≤ G.
VI.3. PERMUTATION GROUP DECOMPOSITIONS 105 Proof: We will establish the bijection by representing each block system by the block B containing ω. For a block B with ω ∈ B, let StabG(B) be the set-wise stabilizer of B. We have that S ≤ StabG(B), as S maps ω to ω and thus must fix the block B.
Since G is transitive on Ω there are elements in G that map ω to an arbitary δ ∈ B, as B is a block this means that these elements must lie in StabG(B). This shows that StabG(B) acts transitively on B and that B = ωStabG(B). The map from blocks to subgroups therefore is injective.
Vice versa, for a subgroup U ≥ S, let B = ωU. Clearly B is, as a set, stabilized by U. We note that in fact U = StabG(B), as any element x ∈ StabG(B) must map ω to ωxinB and by definition there is u ∈ U such that ω = (ωx)u = ωx u, so xu ∈ S ≤ U and therefore x ∈ U . Thus the map from subgroups containing S to subsets containing ω is injective.
We claim that B is the block in a block system. Since G is transitive, the images of B clearly cover all of Ω. We just need to show that they form a partition. For that, assume that Bg∩ Bh /= ∅, that is there exists δ ∈ Bg∩ Bh. This means that δg−1, δh−1 ∈ B = ωU and thus there exists ug, uh ∈ U such that ωugg = δ = ωuhh. But then ugg(uhh)−1 = ugg h−1u−1h ∈ StabG(ω) ≤ U, the gh−1 ∈ U and (as U = StabG(B)) we have that Bg = Bh. This shows that the images of B form a partition of Ω.
The properties shown together establish the bijection. ◻
Corollary VI.14: The blocks systems for G on Ω form a lattice under the “subset”
(that is blocks are either subsets of each other or intersect trivially) relation. Its maximal element is B∞, its minimal element is B0.
We call a transitive permutation group imprimitive, if it affords a nontrivial block system on Ω. We want to ontain an embedding theorem for imprimitive groups, similar to what we did for direct products. That is, we want to describe a “universal” group into which imprimitive groups embed.
The construction for this is called the wreath product”
Definition VI.15: Let A be a group, b an integer, and B ≤ Sba permutation group.
The wreath product A ≀ B is the semidirect product of N = Ab = A × ⋯ × A
´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶
b
with B where B acting on N by permuting the components.
If A ≤ Sais also a permutation group, we can represent A ≀ B as an imprimitive group on a ⋅ b points: Consider the numbers 1, . . . , ab, arranged as in the following diagram.
1 2 ⋯ a
a + 1 a + 2 ⋯ 2a
⋮ a(b − 1) + 1 a(b − 1) + 2 ⋯ ab
Then the direct product N = Abcan be represented as permutations of these
num-bers, the i-th copy of A acting on the i-th row. Now represent the permutations of B by acting simultaneously on the columns, permuting the b rows. The resulting group W is the wreath product A ≀ B. Points in the same row of the diagram will be mapped by W to points in the same row, thus W acts imprimitively on 1, . . . , ab with blocks according to the rows of the diagram. W is therefore called the imprim-itive action of the wreath product.
Lemma VI.16: Let G act imprimitively on 1, . . . n = ab with b blocks of size a. Then G can be embedded into a wreath product Sa≀ Sbin
Proof: By renumbering the points we may assume without loss of generality that the blocks of G are exactly {1, . . . , a}, {a + 1, . . . , 2a} and so on, as given by the rows of the above diagram. Let g ∈ G. Then g will permute the blocks according to a permutation b ∈ Sb. By considering Sbas embedded into Sa≀ Sb, we have that g/b fixes all rows of the diagram as sets and thus is in N = (Sa)b. ◻ (Again, it is possible — for example under the name of induced representations — to give a better description, reducing to a wreath product of the block stabilizers action on its block with the groups action on all the blocks, but doing so requires a bit more work.)